Antipodes

The words "antipodes" and "antipodal" are usually used in the context
of a sphere. Their meaning there is clear: two points in a sphere are
antipodal if the distance between them is as great as possible.

This works because there is an obvious distance measure to use in a
sphere: great circle distance. It is possible to devise at least two
different sensible distance measures in a torus (or on "different
toruses", if you prefer). For surfaces of genus greater than 1, it is
even less clear what distance measure should be used.

In these pages I use these words "antipodes" and "antipodal" in a way
not based on a distance measure. My definition applies to all finite
2-manifolds including the sphere. It is
If a symmetry operation on a regular map (which I
prefer to regard as a polyhedron) fixes some structure (e.g.
a vertex, edge, or face) of it, and also fixes some other structure,
these two structures are said to be antipodal.
This relation is transitive, so sets of two or more structures are
mutually antipodal.

Examples from genus 0

For the sphere, this definition is consistent with the usual one. We
find that the five Platonic solids have the following sets of antipodes

tetrahedron: (vertex, face), (edge, edge).

octahedron: (vertex, vertex), (face, face), (edge, edge).

cube: (vertex, vertex), (face, face), (edge, edge).

icosahedron: (vertex, vertex), (face, face), (edge, edge).

dodecahedron: (vertex, vertex), (face, face), (edge, edge).

just as with the usual definition.

The definition covers structures other than vertices, edges, and
faces; e.g.holes and
Petrie polygons. Here is how
this applies to the Platonic solids.

An example from genus 1

One of the more interesting regular maps on the torus consists of
seven hexagons, each bordering the other
six. It has 14 vertices and 21 edges. Its sets of antipodes are
(vertex, vertex), (face), (edge, edge, edge).
Its rotational symmetry group is C7⋊C6, the Frobenius group of order 42.
It has as its Sylow-subgroups

1

normal

C7,

so its

3

14-sided Petrie polygons

form 1 antipodal

threesome

7

conjugate

C3s,

so its

14

vertices

form 7 antipodal

pairs

7

conjugate

C2s,

so its

21

edges

form 7 antipodal

threesomes

Its Petrie polygons encompass all 14 of its vertices, and are fixed by everything.

An example from genus 2

The genus-2 regular map G2:{8,3} has
six octagonal faces, 16 3-valent vertices, and 24 edges. Its sets
of antipodes are
(vertex, vertex, vertex, vertex, Petrie polygon), (face, face), (edge, edge).
Its rotational symmetry group is GL(2,3). GL(2,3) has eight elements of order 3, so its
Sylow-3-subgroups are